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{{MSC|03A15|28A33}}
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{{MSC|03E15|28A05}}
 
[[Category:Descriptive set theory]]
 
[[Category:Descriptive set theory]]
 
[[Category:Classical measure theory]]
 
[[Category:Classical measure theory]]
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====Algebra of sets====
 
====Algebra of sets====
A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed under the set-theoretic operations of union, intersection and taking complements, i.e. such that
+
Also called Boolean algebra or [[Field of sets|field of sets]] by some authors. A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that
 
* $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
 
* $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
 
* $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
 
* $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
* $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$.
+
* $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$
Indeed it is sufficient to assume that $\mathcal{A}$ satisfies the first two properties to conclude that also
+
(see Section 4 of {{Cite|Ha}}). Indeed it is sufficient to assume that $\mathcal{A}$ satisfies the first two properties to conclude that also the third holds.
the third holds.  
+
 
 +
It follows easily that an algebra is also closed under the operation of taking differences. Algebras are special classes of [[Ring of sets|rings of sets]] (also called Boolean rings). A ring of sets is a nonempty collection $\mathcal{R}$ of subsets of some set $X$ which is closed under the set-theoretic operations of finite union and difference. An algebra can be characterized as a ring containing the set $X$.  
  
 
The algebra generated by a family $\mathcal{B}$ of subsets of $X$ is defined as the smallest algebra $\mathcal{A}$ of subsets
 
The algebra generated by a family $\mathcal{B}$ of subsets of $X$ is defined as the smallest algebra $\mathcal{A}$ of subsets
of $X$ containing $\mathcal{B}$. A simple inductive procedure allows to "construct" $\mathcal{A}$ as follows. $\mathcal{A}_0$
+
of $X$ containing $\mathcal{B}$. A simple procedure to construct $\mathcal{A}$ is the following. Define $\mathcal{A}_0$
consists of all elements of $\mathcal{B}$ and their complements. For any
+
as the set of all elements of $\mathcal{B}$ and their complements. Define $\mathcal{A}_1$ as the elements which are intersections of finitely many elements of $\mathcal{A}_0$. $\mathcal{A}$ consists then of finite unions of arbitrary elements of $\mathcal{A}_1$.
$n\in\mathbb N\setminus \{0\}$ we define $\mathcal{A}_n$ as the collection of those sets which are finite unions or finite
+
 
intersections of elements of $\mathcal{A}_{n-1}$. Then $\mathcal{A}=\bigcup_{n\in\mathbb N} \mathcal{A}_n$.  
+
{{Anchor|sigma-algebra}}
  
 
====$\sigma$-Algebra====
 
====$\sigma$-Algebra====
An algebra of sets that is also closed under countable unions. As a corollary a $\sigma$-algebra is also closed
+
 
 +
An algebra of sets that is also closed under countable unions, cp. with Section 40 of {{Cite|Ha}} (also called Boolean $\sigma$-algebra or $\sigma$-field). Analogously one defines [[Ring of sets|$\sigma$-rings]] (also called Boolean $\sigma$-rings) as rings of sets which are closed under countable unions (see Section 40 of {{Cite|Ha}}). $\sigma$-algebras of $X$ can be characterized as $\sigma$-rings which contain $X$.
 +
 
 +
As a corollary a $\sigma$-algebra is also closed
 
under countable intersections. As above, given a collection $\mathcal{B}$ of subsets of $X$, the $\sigma$-algebra generated
 
under countable intersections. As above, given a collection $\mathcal{B}$ of subsets of $X$, the $\sigma$-algebra generated
by $\mathcal{B}$ is defined as the smallest $\sigma$-algebra of subsets of $X$ containing $\mathcal{B}$. The explicit
+
by $\mathcal{B}$ is defined as the smallest $\sigma$-algebra of subsets of $X$ containing $\mathcal{B}$ (also called [[Borel field of sets|Borel field]] generated by $\mathcal{B}$). A
construction given above for the algebra generated by $\mathcal{B}$ can be extended to $\sigma$-algebras with the aid of
+
construction can be given using
 
[[Transfinite number|transfinite numbers]]. As above, $\mathcal{A}_0$consists of all elements of $\mathcal{B}$ and their complements.
 
[[Transfinite number|transfinite numbers]]. As above, $\mathcal{A}_0$consists of all elements of $\mathcal{B}$ and their complements.
 
Given a countable ordinal $\alpha$, $\mathcal{A}_\alpha$ consists of those sets which are countable unions or countable intersections
 
Given a countable ordinal $\alpha$, $\mathcal{A}_\alpha$ consists of those sets which are countable unions or countable intersections
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\bigcup_{\beta<\alpha} \mathcal{A}_\beta\, .
 
\bigcup_{\beta<\alpha} \mathcal{A}_\beta\, .
 
\]
 
\]
$\mathcal{A}$ is the union of the classes $\mathcal{A}_\alpha$ where the index $\alpha$ runs over all countable ordinals.  
+
$\mathcal{A}$ is the union of the classes $\mathcal{A}_\alpha$ where the index $\alpha$ runs over all countable ordinals
 +
(cp. with Exercise 9 of Section 5 in {{Cite|Ha}} where the same construction is outlined for $\sigma$-rings).
  
 
====Relations to measure theory====
 
====Relations to measure theory====
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Therefore $\sigma$-algebras play a central role in measure theory, see for instance [[Measure space]].
 
Therefore $\sigma$-algebras play a central role in measure theory, see for instance [[Measure space]].
  
According to the theorem of extension of measures, any $\sigma$-finite, $\sigma$-additive measure,  defined on an algebra A, can be uniquely  extended to a $\sigma$-additive measure  defined on the $\sigma$-algebra generated  by $A$.
+
According to the theorem of extension of measures, any $\sigma$-finite, $\sigma$-additive measure,  defined on an algebra $A$, can be uniquely  extended to a $\sigma$-additive measure  defined on the $\sigma$-algebra generated  by $A$.
  
 
====Examples.====
 
====Examples.====
1) Let $X$ be an arbitrary set. The collection of finite subsets of $X$ and their complements is an algebra of sets. The collection of subsets
+
1) Let $X$ be an arbitrary set. The collection of finite subsets of $X$ and their complements is an algebra of sets (so-called finite-cofinite algebra). The collection of subsets
of $X$ which are at most countable and of their complements is a $\sigma$-algebra.  
+
of $X$ which are at most countable and of their complements is a $\sigma$-algebra (so-called countable-cocountable σ-algebra).  
  
 
2) The collection of finite unions of intervals of the type
 
2) The collection of finite unions of intervals of the type
 
\[
 
\[
\{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where $-\infty \leq a <b\leq \infty$}
+
\{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where } -\infty \leq a <b\leq \infty
 
\]
 
\]
 
is an algebra.
 
is an algebra.
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3) If $X$ is a topological space, the elements of the $\sigma$-algebra generated by the open sets are called [[Borel set|Borel sets]].
 
3) If $X$ is a topological space, the elements of the $\sigma$-algebra generated by the open sets are called [[Borel set|Borel sets]].
  
4) The Lebesgue measurable sets of $\mathbb R^k$ form a $\sigma$ algebra (see [[Lebesgue measure]]).
+
4) The Lebesgue measurable sets of $\mathbb R^k$ form a $\sigma$-algebra (so-called Lebesgue σ-algebra, see [[Lebesgue measure]]).
  
 
5) Let $T$ be an arbitrary set and consider $X = \mathbb R^T$ (i.e. the set of all real-valued functions on $\mathbb R$).
 
5) Let $T$ be an arbitrary set and consider $X = \mathbb R^T$ (i.e. the set of all real-valued functions on $\mathbb R$).
 
Let $A$ be the class of sets of the type
 
Let $A$ be the class of sets of the type
 
\[
 
\[
\{\omega\in \mathbb R^T: (\omega (t_1), \ldots,\omega t_k)\in E\}
+
\{\omega\in \mathbb R^T: (\omega (t_1), \ldots,\omega (t_k))\in E\}
 
\]
 
\]
 
where $k$ is an arbitrary natural number, $E$ an arbitrary Borel subset of $\mathbb R^k$ and $t_1,\ldots, t_k$
 
where $k$ is an arbitrary natural number, $E$ an arbitrary Borel subset of $\mathbb R^k$ and $t_1,\ldots, t_k$
an arbitrary collection of distinct elements of $T$. $A$ is an algebra of subsets of $\mathbb R^T$.
+
an arbitrary collection of distinct elements of $T$. $A$ is an algebra of subsets of $\mathbb R^T$ (so-called cylindrical algebra).
 
In the theory of random processes a [[Probability measure|probability measure]]
 
In the theory of random processes a [[Probability measure|probability measure]]
 
is often originally defined only on an algebra of this type, and then subsequently extended to the $\sigma$-algebra generated by $A$.
 
is often originally defined only on an algebra of this type, and then subsequently extended to the $\sigma$-algebra generated by $A$.
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====References====
 
====References====
 
{|
 
{|
|valign="top"|{{Ref|Bo}}||      N. Bourbaki, "Elements of mathematics. Integration" ,  Addison-Wesley    (1975) pp. Chapt.6;7;8 (Translated from French)  {{MR|0583191}}    {{ZBL|1116.28002}} {{ZBL|1106.46005}}  {{ZBL|1106.46006}}    {{ZBL|1182.28002}} {{ZBL|1182.28001}}  {{ZBL|1095.28002}}    {{ZBL|1095.28001}} {{ZBL|0156.06001}}
+
|valign="top"|{{Ref|Bo}}||      N. Bourbaki, "Elements of mathematics. Integration",  Addison-Wesley    (1975) pp. Chapt.6;7;8 (Translated from French)  {{MR|0583191}}    {{ZBL|1116.28002}} {{ZBL|1106.46005}}  {{ZBL|1106.46006}}    {{ZBL|1182.28002}} {{ZBL|1182.28001}}  {{ZBL|1095.28002}}    {{ZBL|1095.28001}} {{ZBL|0156.06001}}
 
|-
 
|-
|valign="top"|{{Ref|DS}}||    N. Dunford, J.T. Schwartz, "Linear  operators. General theory" ,  '''1''' , Interscience (1958)  {{MR|0117523}}
+
|valign="top"|{{Ref|DS}}||    N. Dunford, J.T. Schwartz, "Linear  operators. General theory",  '''1''', Interscience (1958)  {{MR|0117523}} {{ZBL|0635.47001}}
 
|-
 
|-
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory" , v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
+
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 
|-
 
|-
|valign="top"|{{Ref|Ne}}||  J. Neveu,   "Bases mathématiques du calcul des probabilités" , Masson (1970)
+
|valign="top"|{{Ref|Ne}}||  J. Neveu, "Mathematical foundations of the calculus of probability", Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam 1965 {{MR|0198505}} {{ZBL|0137.1130}}
 
|-
 
|-
 
|}
 
|}

Latest revision as of 09:36, 16 August 2013

2020 Mathematics Subject Classification: Primary: 03E15 Secondary: 28A05 [MSN][ZBL]

Algebra of sets

Also called Boolean algebra or field of sets by some authors. A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that

  • $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
  • $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
  • $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$

(see Section 4 of [Ha]). Indeed it is sufficient to assume that $\mathcal{A}$ satisfies the first two properties to conclude that also the third holds.

It follows easily that an algebra is also closed under the operation of taking differences. Algebras are special classes of rings of sets (also called Boolean rings). A ring of sets is a nonempty collection $\mathcal{R}$ of subsets of some set $X$ which is closed under the set-theoretic operations of finite union and difference. An algebra can be characterized as a ring containing the set $X$.

The algebra generated by a family $\mathcal{B}$ of subsets of $X$ is defined as the smallest algebra $\mathcal{A}$ of subsets of $X$ containing $\mathcal{B}$. A simple procedure to construct $\mathcal{A}$ is the following. Define $\mathcal{A}_0$ as the set of all elements of $\mathcal{B}$ and their complements. Define $\mathcal{A}_1$ as the elements which are intersections of finitely many elements of $\mathcal{A}_0$. $\mathcal{A}$ consists then of finite unions of arbitrary elements of $\mathcal{A}_1$.

$\sigma$-Algebra

An algebra of sets that is also closed under countable unions, cp. with Section 40 of [Ha] (also called Boolean $\sigma$-algebra or $\sigma$-field). Analogously one defines $\sigma$-rings (also called Boolean $\sigma$-rings) as rings of sets which are closed under countable unions (see Section 40 of [Ha]). $\sigma$-algebras of $X$ can be characterized as $\sigma$-rings which contain $X$.

As a corollary a $\sigma$-algebra is also closed under countable intersections. As above, given a collection $\mathcal{B}$ of subsets of $X$, the $\sigma$-algebra generated by $\mathcal{B}$ is defined as the smallest $\sigma$-algebra of subsets of $X$ containing $\mathcal{B}$ (also called Borel field generated by $\mathcal{B}$). A construction can be given using transfinite numbers. As above, $\mathcal{A}_0$consists of all elements of $\mathcal{B}$ and their complements. Given a countable ordinal $\alpha$, $\mathcal{A}_\alpha$ consists of those sets which are countable unions or countable intersections of elements belonging to \[ \bigcup_{\beta<\alpha} \mathcal{A}_\beta\, . \] $\mathcal{A}$ is the union of the classes $\mathcal{A}_\alpha$ where the index $\alpha$ runs over all countable ordinals (cp. with Exercise 9 of Section 5 in [Ha] where the same construction is outlined for $\sigma$-rings).

Relations to measure theory

Algebras (respectively $\sigma$-algebras) are the natural domain of definition of finitely-additive ($\sigma$-additive) measures. Therefore $\sigma$-algebras play a central role in measure theory, see for instance Measure space.

According to the theorem of extension of measures, any $\sigma$-finite, $\sigma$-additive measure, defined on an algebra $A$, can be uniquely extended to a $\sigma$-additive measure defined on the $\sigma$-algebra generated by $A$.

Examples.

1) Let $X$ be an arbitrary set. The collection of finite subsets of $X$ and their complements is an algebra of sets (so-called finite-cofinite algebra). The collection of subsets of $X$ which are at most countable and of their complements is a $\sigma$-algebra (so-called countable-cocountable σ-algebra).

2) The collection of finite unions of intervals of the type \[ \{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where } -\infty \leq a <b\leq \infty \] is an algebra.

3) If $X$ is a topological space, the elements of the $\sigma$-algebra generated by the open sets are called Borel sets.

4) The Lebesgue measurable sets of $\mathbb R^k$ form a $\sigma$-algebra (so-called Lebesgue σ-algebra, see Lebesgue measure).

5) Let $T$ be an arbitrary set and consider $X = \mathbb R^T$ (i.e. the set of all real-valued functions on $\mathbb R$). Let $A$ be the class of sets of the type \[ \{\omega\in \mathbb R^T: (\omega (t_1), \ldots,\omega (t_k))\in E\} \] where $k$ is an arbitrary natural number, $E$ an arbitrary Borel subset of $\mathbb R^k$ and $t_1,\ldots, t_k$ an arbitrary collection of distinct elements of $T$. $A$ is an algebra of subsets of $\mathbb R^T$ (so-called cylindrical algebra). In the theory of random processes a probability measure is often originally defined only on an algebra of this type, and then subsequently extended to the $\sigma$-algebra generated by $A$.

References

[Bo] N. Bourbaki, "Elements of mathematics. Integration", Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958) MR0117523 Zbl 0635.47001
[Ha] P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Ne] J. Neveu, "Mathematical foundations of the calculus of probability", Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam 1965 MR0198505 Zbl 0137.1130
How to Cite This Entry:
Algebra of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebra_of_sets&oldid=27294
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article